極限

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1 si θ = ) θ 0 θ cos θ θ 0 θ = ) P T θ H A, 0) θ, 0 < θ < π ) AP, P H A P T PH < AP < AT si θ < θ < ta θ si θ < θ < si θ cos θ θ cos θ < si θ θ < θ < 0 θ = h θ 0 cos θ =, θ 0 si θ θ =. θ 0 cos θ θ θ 0 cos θ) + cos θ) θ + cos θ) θ 0 si ) θ si θ θ + cos θ) θ 0 θ + cos θ) = ), θ 0 0 ) si θ si θ θ θ cos θ θ cos θ θ cos θ θ

2 . i) θ = π { si θ = si π = = 0.5 θ = π =.459 si θ θ 0.5 = cos θ = cos π = = = θ = ).459 = ) = cos θ θ θ = π θ ii)θ = π si θ = si π = si45 0 ) = 4 = θ = π =.459 = si θ θ cos θ = cos π = + 4 = = θ = ).459 = ) = cos θ θ θ si θ θ cos θ θ = = si π π π π = cos = = si, = = cos = = = cos 0, ) = cos.)

3 ) 0 si si ) 0 si si ) ) 0 cos 4) 0 cos cos

4 4 = = e = 4 = =.5 A0, ) = a A0, ) A = a a A0, ) a e ) f) = e f fh) f0) e 0) h 0 h h h 0 h = e A0, ) h 0 e h h = e ) = e f ) h 0 e +h e h h 0 e h h e = e Ph, e h ) e ) = e A0, ) h e h AP eh h g) = a h 0 A g ) h 0 a +h a h h 0 a h h a ) = h, 4 ) = 4 4 h h 0 h h 0 h a = e log e a = e log a h 0 a h h h 0 e h loge a h h 0 e h loge a h log e a a ) = a log a. log e a = log e a ) e = e.

5 5 e 4 ) 0 e = e =t log+t)= t 0 log + t) t = t 0 + t) t = e ± + ) = e 0 e e +. log + ) log + ) = +, = = e, = log + ) 0, ), 0, 0) = log + ), = = log, = = = log, 0) = e = log = = = = e = + = = log + ) = = log = log ) = t log + t) t 0 t,, ) 4 = log + t) t = = log e + t) t = e t 0 t 0 + ) = e e A

6 = e = log log = = log = e a e a. a = XX > 0) = log X e a = e e a e a = log ) = > 0) = log a e a -. ) 0 e ) 0 log + ) ) 0 5 4) 0 e si log + ) 5) 0 logcos )

7 7. e ) e ; e + ).) a = ) + =,,, ). a = + =, a = + ) = 9 4 =.5, a = + ) = 4 7 =.70707, a 4 = + 4 ) 4 = 5 5 =.44405, a 00% ) )? ) + ) = + ) : + ) = 4. = : + ) = + ) ). + : + ) + ) = + ). 4, + ) = a ) = + a a 0., 0.0, ), a + ) = e ) + ) + ) + ) e e ) ) 5 5 ), )

8 8 P, + ), P, + ) ), P, + ) ) k,, P k, + ) ) k,, P k k =,,, ) = ) + = { +, ) } f ) = { + ) } { f ) + ) } = e ) = f ) = e 4) ) + = + ) e ) + = e = + ) = + = + P P P P A0, ) A0, ) ) - + = e. 5) ) + ) ) + ) ) + ) 4) P, + ) A0, ) AP P = +.) P = e Q, e ) Q A, = + A = e 5) ),) 00% ) 00%.

9 9. ) + ) ) ) 0 4) 5) ) + + ) ) + ) ) 7) = log = e,, =, =,, = < < < < < e) = log a =, b = e b a e a b

10 0 4 ) ) f), g) = a fa) = ga) = 0, = a g )=\ 0 f ) a g ) a f) g) a f ) g ) g a)=\ 0 a f) g) a f) fa) a g) ga) a = f a) g a) 0 0 ) : ) log cos ) 0 ) si ) f) =, g) = log f) = g) = 0. g ) =, g )=\ 0. log f) f) g) g) = f ) g ) = = f ) =, g ) = ) f) = cos, g) = f0) = g0) = 0 g ) = g 0) = 0. g ) = 0 = 0, 0 cos cos ) si 0 ) 0 = 0 si = ) f) = si, g) = f0) = g0) = 0 g ) = g ) = 0 = 0 si si ) cos 0 ) 0 cos ) si 0 ) 0 = ) 0 0 ) ) =. 0 )

11 4 ) 0 ) 0 cos XY Pt, cos t) t t = 0 P C =\ 0 A, cos ), A cos = A ) C A Q Q C A ) Q t = c 0 < c < < c < 0), Q C f c) g c) A ) cos = f c) g c) 0 < c < < c < 0) c 0 ) 0 c 0. c 0 f c) g c) 0 cos c 0 f c) g c) c 0 si c c t 0 si t t = t = c Y Y t = c A, cos ) Q C X 0 A, cos ) Q C X ) ) t = 0 g t) = 0 C : f c) g c) t = 0 g t) = 0

12 4 ) ) a f) g) Y { X = gt) C : Y = ft) t = a C :gt), ft)) Ag), f)) =\ a g )=\ 0 = a = a g)=\ 0. ga), fa)) = 0, 0), f) g) = A ) A Q t = t = c t = a X C A Q Q C A Q t = c 0 < c < < c < 0), Q C f c) g c) A ) f) g) = f c) g c) < c < < c < ) f c. a c a c) c a g c) a f) g) c a f c) g c) a f ) g ) Q.E.D. 4.) ) 0 e + ) ) 0 e + e ).

13 4 ) 4, ) ) 0 e + ) {e + )} 0 ) 0 e = ) 0 e + e {e + e } e e 0 ) 0 0 {e e } ) 0 e + e = 0 e + e 0 e + e e e ) 0 e = Commet ) ) 0 e + ) e + + e 0. ) ) e + e + ) 0 e + e = + ) + ) = 0 = 0 ) e + + e + e 0 ) ) ) = =.K.) 0 0,,, 0 0 MuPAD.)

14 4 5 Talor ) = a f ) a) = a f) = fa) + f a) a) + f a)! =,, a) + f a)! a) + + f ) a)! a) [ ] f) = fa) + f a) a) [ ] f) = fa) + f a) a) + f a)! [ ] f) = fa) + f a) a) + f a)! a) a) + f a)! a) ) a = 0, = 0 f ) 0) = 0 =,, f) = f0) + f 0) + f 0)! + f 0)! + + f ) 0)! [ ] f) = f0) + f 0) [ ] f) = f0) + f 0) + f 0)! [ ] f) = f0) + f 0) + f 0)! + f 0)! ) f ) a) f ) a) = f a), f ) a) = f a), f ) a) = f a). a).

15 5 Talor ) 5 5. f) = a a f) fa) a = f a) Q h) = f) g) = f) A P a f) fa) f a) a) a = 0 g) = f a) a) + fa) = a! ) = g) a a a f) g) a = 0 f) g) h) a a) = a = g) = f), f) = a = a f) fa) + a)f a) f) h) = f) {fa) + a)f a)} a ). ) ) a α, β 0 β a α = 0 β α β α ). β a α β α = k k )

16 5 Talor ) a) 4 0m 5m,.5m.) =, =. h),, ) i) f) = a, a h) = f) g) < = f) = g).. Q = f) = g) Q Q P A P P a 4 ii) f) = a f) = a = a Aa, 0) = 0 a+0 f) fa) a =, a 0 f) fa) a =. f f) fa) a) a a f) = a = a = a = 0 = a = = 0 = a Q = a Q = a Q Q Q Aa, 0) P P P Q Aa, 0) P P Aa, 0) P 4 = f a) a) + fa) = a = f).

17 5 Talor ) 7 5. = 0 = 0 f ) 0) = 0 f) = f0) + f 0) + f 0)! + f 0)! + + f ) 0)! 5.. e = 0 f) = e f ) = e, f ) = e, f ) = e f0) = f 0) = f 0) = f 0) = e + +! +! = 0 = f) ) = + f) A A 0 e + ) {e + )} e 0 ) 0 = 0 e = = 0 e + ) e + + [ ] f) B ) e + + B 0 0 {e + )} e 0 ) 0 { )} e + + e + ) ) 0 = = 0 ) e + + e [ ]

18 5 Talor ) 8 0 a + b) ±., e + ) a = e = + = e + ) = = e + ) = = e + ) 0 a + b + c) ±, = e = + + ) = e + + ) = e + + = e : + + a = e + + ) = = e 0,, = = + + = + = e e = 0,,

19 5 Talor ) si = 0 f) = si f ) = cos, f ) = si, f ) = cos f0) = 0, f 0) =, f 0) = 0, f 0) = si 0 +! + 0! +! si = 0 = f) ) = f) A, A 0 si si ) cos 0 ) 0 0 cos ) ) 0 si = 0, si + 0 = f) + 0 B, B 0 si si ) cos 0 ) 0 0 cos ) si 0 ) 0 = si si [ ] = = si = si = si = si : + 0 a

20 5 Talor ) 0 = si, = = = si = = si = si = 0,, 5.. cos = 0 f) = cos f ) = si, f ) = cos, f ) = si f0) =, f 0) = 0, f 0) =, f 0) = 0 cos + 0! +! + 0! cos = 0 = f) ) = f) A,, A 0 cos = cos [ ]

21 5 Talor ) A a f) {fa) + f a) a)} a) F ), G), F a) = Ga) = 0. f a) A a f) {fa) + f a) a)} a) a f) {fa) + f a) a)}) a) ) = 0 f) a f ) f a) a) = f a) f) fa) + f a) a) + f a) f a), = a f) a). f) fa) + f a) a) + f a) a) a) f a) a f ) f a) a 5.. f a), f) B a a { fa) + f a) a) + f a) a) } a) 0 0 { f) fa) f a) a) f a) a) } { a) } a f ) f a) f a) a) a) 0 0 a {f ) f a) f a) a)} { a) } a f ) f a) a) = f a) { } f) fa) + f a) a) + f a) a) a) f a) f a) a f ) f a) a

22 5 Talor ) a f a) f) fa) + f a) a) + f a) f) = fa) + f a) a) + f a)! a). a) + f a) a) + f a)! a) a) 5.4., = 0 = 0 e + +! si! cos! +! + 5 5! + 4 4! + +! 7 7!! log + ) ) α + α! + αα )! + + ) )! + + ) )! + + ) + αα )α )! + + αα ) α ))! α. α α = ) ) + )! ) ) + )α α =, = ) = = + +, 0 + = 0.

23 5 Talor ) si si. ) si = 0! = π si π π π ) ) si π = 0.5, π.459 = log a a a??) ) log + ) = ) log ) = log + ) = ) 0 < < + + = = { ) log + ) } = { 5 ) log + ) + ) } 5 = log = e = = + + =.5 e + +! +! + 4! + +! = = = = = e.78888, +)! = 7 8! = ) + e

24 4. e e = 0 e =t t 0 log + t) t = t 0 + t) t = e ± + ) = e ) e 0 = e + e e.)?) a = ) + = ±, ±, ±, ). a = + =, a = + ) = 9 4 =.5, a = + ) = 4 7 =.707, a4 = + 4 ) 4 = 5 5 =.4440, + ) ) ) ) = C 0 + C + C + + C = + + )! = + + )! ) ) +! + ) )! ) + +! + + )! a + ) = + + ) + ) ) + + )!!! < + +! +! + +! < ! = 4 > = = + ) < a e ) e + ) = e ) ) a a a e )

25 5. ) 0 si si si 0 si = ) 0 +0, cos, 0 cos 4) si si ) ) 0 = 0 si si ) si si = cos cos = cos + ) cos ) α ± β = cos cos si si ) cos si + si si ) = si si Commet cos cos 0 0 si si si θ θ, cos θ θ 0 4) si si = 4 ) 0 ) 0 si si = ) si si ) si ) = cos = ) = 9, cos cos cos 9 ) ) = 4 = 4 - ) ) 5 ) 0 4) 0 e log 5) 0 e log 5) log 5) log 5 = log 5 0 e si log + ) e si 0 si si log + ) = = 5) logcos ) 0 Commet 0 log + cos ) 0 log + cos ) cos 0 e +, log + ) 4) 0 5) 0 e si log + ) = logcos ) = log ). cos ) = ) = logcos ) =

26 - ) { + ) } = e ) ) + ) { + Commet ) } = e { + ) } = e ) 00% e ) 00% e ) 00% e. ) ) ) 4) 5) ) + + ) ) + + ) + + ) ) = ) + + ) ) ) + { = + ) = + + ) + ) ) = ) + ) = ) } ) + ) < + < ) < + < =, ) + = + =

27 7 7) = log, 0), = S a = T log e =, b S a, T b S = T a b. b a = S a = e b T = log A, 0), P + a, 0), Q + a, log + a )), R + a, a ) = log A = log S = log = = APQ < S < APR a log + a ) < S < a = T b loge b ) < T < b = = log R Q + a, log + a )) S = T, a log + a ) < S = T < b b loge b ) < T = S < a A, 0) a = P + a, 0) = log log + a ) < b a a < loge b ) loge b ) Se b, 0) b e a 0, b 0 log + a ) a t 0 log + t) t =, loge b ) = log e = Commet b a = S T log e = =\ 0 T log = 0 S

28 8 ) ) ) 4) iteret ), MuPAD MuPAD free,.) ) geogebra free MuPAD ) cabri geogebra 4) Mathematica CATComputer algebra sstem) free Mathematica ) si 0 π) si ) plot plot si, =0..pi = si 0 π) d si d/d d/d si d π it it si), =0..pi si d 0 si si )/ as ->0 0 * ) *si si si si / si )/ ^ ^ log log π e, π e, pi it si, =0..pi si d 0 plot ^- =

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1 1 1 7 1.1.................................. 11 2 13 2.1............................ 13 2.2............................ 17 2.3.................................. 19 3 21 3.1.............................

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